A210731 a(n) = a(n-1) + a(n-2) + n + 3 with n>1, a(0) = a(1) = 0.

0, 0, 5, 11, 23, 42, 74, 126, 211, 349, 573, 936, 1524, 2476, 4017, 6511, 10547, 17078, 27646, 44746, 72415, 117185, 189625, 306836, 496488, 803352, 1299869, 2103251, 3403151, 5506434, 8909618, 14416086, 23325739, 37741861, 61067637, 98809536

Offset: 0

Author

Alex Ratushnyak, May 10 2012

Keywords

• nonn
• easy

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-1,1).

Crossrefs

Cf. A033818: a(n)=a(n-1)+a(n-2)+n-5, a(0)=a(1)=0 (except first 2 terms and sign).

Cf. A002062: a(n)=a(n-1)+a(n-2)+n-4, a(0)=a(1)=0 (except the first term and sign).

Cf. A065220: a(n)=a(n-1)+a(n-2)+n-3, a(0)=a(1)=0.

Cf. A001924: a(n)=a(n-1)+a(n-2)+n-1, a(0)=a(1)=0 (except the first term).

Cf. A023548: a(n)=a(n-1)+a(n-2)+n, a(0)=a(1)=0 (except first 2 terms).

Cf. A023552: a(n)=a(n-1)+a(n-2)+n+1, a(0)=a(1)=0 (except first 2 terms).

Cf. A210730: a(n)=a(n-1)+a(n-2)+n+2, a(0)=a(1)=0.

Programs

GAP

F:=Fibonacci;; List([0..40], n-> F(n+3)+4*F(n+1)-n-6); # G. C. Greubel, Jul 09 2019

Magma

F:=Fibonacci; [F(n+3)+4*F(n+1)-n-6: n in [0..40]]; // G. C. Greubel, Jul 09 2019

Mathematica

With[{F = Fibonacci}, Table[F[n+3]+4*F[n+1]-n-6, {n,0,40}]] (* G. C. Greubel, Jul 09 2019 *)
nxt[{n_,a_,b_}]:={n+1,b,a+b+n+4}; NestList[nxt,{1,0,0},40][[;;,2]] (* or *) LinearRecurrence[{3,-2,-1,1},{0,0,5,11},40] (* Harvey P. Dale, Dec 30 2024 *)

PARI

vector(40, n, n--; f=fibonacci; f(n+3)+4*f(n+1)-n-6) \\ G. C. Greubel, Jul 09 2019

Sage

f=fibonacci; [f(n+3)+4*f(n+1)-n-6 for n in (0..40)] # G. C. Greubel, Jul 09 2019

Formula

From Colin Barker, Jun 29 2012: (Start)

a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).

G.f.: x^2*(5-4*x)/((1-x)^2*(1-x-x^2)). (End)

a(n) = Fibonacci(n+3) + 4*Fibonacci(n+1) - (n+6). - G. C. Greubel, Jul 09 2019